Chapter 11, Problem 35P

(a)

To determine

To Find: The position of pendulum.

Answer to Problem 35P

The position of pendulum at time t = 0.25 seconds is $-11°$ .

Explanation of Solution

Given:

Frequency of oscillation of pendulum, $f=2.5\text{ Hz}$

Angle, ${\theta }_{\max }=15°$

Formula used:

Equation of motion for the object of simple harmonic motion that has maximum displacement at time t = 0 is given by,

  $x=A\times \cos (\omega t)=A\times \cos (2\pi \times f\times t)$

Here,

  $\omega$ is angular frequency of pendulum,

  $A$ is amplitude of pendulum,

  $f$ is frequency of oscillation,

  $t$ is time.

For pendulum $x=L\times \theta$

Here,

  $x$ is linear displacement of pendulum,

  $L$ is length of pendulum,

  $\theta$ is angular displacement of pendulum.

Calculation:

Maximum displacement is given by following formula,

Maximum displacement $x_{\max }=A=L\times {\theta }_{\max }$

  $L\times \theta =L\times {\theta }_{\max }\times \cos (2\pi \times f\times t)$

So,

Angular displacement of the pendulum is,

  $\begin{array}{l}\theta ={\theta }_{\max }\times \cos (2\pi \times f\times t)\\ \theta =15°\times \cos (2\pi \times 2.5\times t)\\ \theta =15°\times \cos (5\pi \times t)\end{array}$

Now, $t=0.25\text{ s}$

  $\begin{array}{l}\theta =15°\times \cos (5\pi \times 0.25)\\ \theta =-11°\end{array}$

Conclusion:

Thus, the position of pendulum at time $t=0.25\text{ s}$ is $-11°$ .

(b)

To determine

To Find: The position of pendulum.

Answer to Problem 35P

The position of pendulum at time $t=\text{ 1}\text{.60 s}$ is $15°$ .

Explanation of Solution

Given:

Frequency of oscillation of pendulum $f=2.5\text{ Hz}$

Angle ${\theta }_{\max }=15°$

Formula used:

Equation of motion for the object of simple harmonic motion that has maximum displacement at time t = 0 is given by,

  $x=A\times \cos (\omega t)=A\times \cos (2\pi \times f\times t)$

Here,

  $A$ is amplitude of pendulum.

  $\omega$ is angular frequency of pendulum,

  $f$ is frequency of oscillation,

  $t$ is time.

For pendulum $x=L\times \theta$

Here,

  $x$ is linear displacement of pendulum,

  $L$ is length of pendulum,

  $\theta$ is angular displacement of pendulum.

Calculation:

Maximum Displacement is given by following formula,

Maximum displacement $x_{\max }=A=L\times {\theta }_{\max }$

  $L\times \theta =L\times {\theta }_{\max }\times \cos (2\pi \times f\times t)$

So,

Angular displacement of the pendulum is,

  $\begin{array}{l}\theta ={\theta }_{\max }\times \cos (2\pi \times f\times t)\\ \theta =15°\times \cos (2\pi \times 2.5\times t)\\ \theta =15°\times \cos (5\pi \times t)\end{array}$

Now, $t=\text{ 1}\text{.60 s}$

  $\begin{array}{l}\theta =15°\times \cos (5\pi \times 1.60)\\ \theta =15°\end{array}$

Conclusion:

Thus, the position of pendulum at time $t=\text{ 1}\text{.60 s}$ is $15°$ .

(c)

To determine

To Find: The position of pendulum.

Answer to Problem 35P

The position of pendulum at time $t=\text{ 500 s}$ is $15°$ .

Explanation of Solution

Frequency of oscillation of pendulum, $f=2.5\text{ Hz}$

Angle, ${\theta }_{\max }=15°$

Formula used:

Equation of motion for the object of simple harmonic motion that has maximum displacement at time t = 0 is given by,

  $x=A\times \cos (\omega t)=A\times \cos (2\pi \times f\times t)$

Here,

  $A$ is amplitude of pendulum.

  $f$ is frequency of oscillation.

  $t$ is time.

For pendulum $x=L\times \theta$

Here,

  $L$ is length of pendulum.

  $\theta$ is angular displacement of pendulum.

Calculation:

Maximum Displacement is given by following formula,

Maximum displacement $x_{\max }=A=L\times {\theta }_{\max }$

  $L\times \theta =L\times {\theta }_{\max }\times \cos (2\pi \times f\times t)$

So,

Angular displacement of the pendulum is,

  $\begin{array}{l}\theta ={\theta }_{\max }\times \cos (2\pi \times f\times t)\\ \theta =15°\times \cos (2\pi \times 2.5\times t)\\ \theta =15°\times \cos (5\pi \times t)\end{array}$

Now, $t=\text{ 500 s}$

  $\begin{array}{l}\theta =15°\times \cos (5\pi \times 500)\\ \theta =15°\end{array}$

Conclusion:

Thus, the position of pendulum at time $t=\text{ 500 s}$ is $15°$ .